On 2/12/2013 9:19 AM, Craig Feinstein wrote:> Let's say I have a drawer of an infinite number of identical socks at time zero. I take out one of the socks at time one. Then the contents of the drawer at time zero is identical to the contents of the drawer at time one, since all of the socks are identical and there are still an infinite number of them in the drawer at both times. But the contents of the drawer at time zero is also identical to the contents of the drawer at time one plus the sock that was taken out, since they are exactly the same material. So we have the equations:>> Contents of drawer at time 0 = Contents of drawer at time 1> Contents of drawer at time 0 = (Contents of drawer at time 1) plus (sock taken out of drawer).>> Subtracting the equations, we get>> Nothing = sock taken out of drawer.>> This is false, so infinity cannot exist.>> How does modern mathematics resolve this paradox?>

Consider your use of the word "identical".

In Aristotle, one finds the discussion that one cannever define x=y. To be precise, he says that onecan never prove a definition, but one can destroya definition. But, definitions rely on the notionthat some word is "the same" as the object towardwhich its language act of referring is directed.

Now, what does Aristotle mean by this? He meansprecisely the kind of thing that Einstein said aboutrelativity. It cannot be proved true, but it canbe proved false by a single experiment.

In finite-state automata -- a finite mathematicaldiscipline -- the definition of distinguishabilityis categorized in terms of experiments of lengthk, where k is the length of a given input string.Two automaton states are k-distinguishable iftheir is an experiment of length k for which theautomaton is started in each of the candidatestates and the two output strings differ. Theyare k-equivalent if their is no m<=k for whichthey are m-distinguishable.

Thus, two states are distinguishable if theyare k-distinguishable for some particular k.

Two states are equivalent if they are notdistinguishable.

How many experiments does it take to provethat they are equivalent?

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Suppose now, that you do not like this explanationbecause you are talking about socks, and, socksare material objects.

How do we understand material objects? We canthink of them as bodies in the sense of impenetrability.We can think of them as bodies in terms of ourvisual field.

Both senses of body are correlated with certainforms of logic. In general, one thinks of deonticlogic in terms of social norms of what is andwhat is not permitted. But, the general senseof deontic logic is simply the logic of lawfulness.Clearly, the notion of a physical law like gravityarises from a "law" like "birds can fly, but mancannot." Or, returning to the impenetrabilityof bodies mentioned above, "one cannot walkthrough walls."

The logic associated with the visual field issomewhat harder. I will argue here that it istemporal logic. I probably cannot do thatsuccessfully because of the complexities. But,our visual field delineates objects on the basisof color distinctions. Our science has investeda great deal of effort to explain color in ourvisual field, and, that has led to optics andquantum mechanics. It has also led to specialrelativity in the sense that the color oflight is presumed to be different to observersin different inertial reference frames.

But, if one grants me this position, then whatone has are two distinct logics whose commonelements form what is called propositionallogic. Typically, what is true and what isfalse in propositional logic is based ona set of functions called basic Boolean functions,or truth tables.

Now, the basic Boolean functions are part ofa class of functions called switching functions.These switching functions can have a propertycalled linear separability. Not all switchingfunctions are linearly separable. But, whatmakes linear separability important is thatit has a geometric analogue.

The sense by which you cannot walk throughwalls is the sense by which one representslinear separability. In your typical introductorymathematics classes, a line divide a plane intotwo halves and a plane divides space into twohalves.

And, now we can return to the question ofyour socks. For how do we think of ourmaterial world as consisting of a pluralityof objects if we do not first divide thefield of sensory experience into parts?

When we divide the world into parts, weare applying the logics mentioned abovein the sense that we are explaining boundaries.

The origin of the mathematical theory of setsarose in combination with the mathematical theoryof point set topology. The two arise togetherbecause they are attempts to address the problemof Xeno's paradox. Calculus and the numericalmethods of approximation arising from the solutionof its equations address Xeno's paradox byquantizing the last step to the finish line.That is, they get informationally "close enough"and then treat the error as a discrete quanta ofnoise. Point set topology addresses the questionof Xeno's paradox as a lawlike limitation relatedto the boundaries of material objects.

Once again, we arrive at the problem ofidentity. In particular, the issue here isLeibniz' principle of identity of indiscernibles.Since the resources available to you willnot explain this as Leibniz actually wroteit, I shall give you the quote:

"What St. Thomas affirms on thispoint about angels or intelligences('that here every object is a lowestspecies') is true of all substances,provided one takes the specific differencein the way that geometers take it withregard to their figures."

In point set topology, this is expressedfor topologies that have a way to measurespecific differences by Cantor's intersectiontheorem. This theorem relates a sequenceof nested non-empty closed sets havingvanishing diameters.

It is in the definition of a closed setwhere one is confronted with the boundarywhere Xeno's paradox comes into play oncemore.

So, now if we return to the discussion oflinear separability, it turns out that theplanes by which we divide our sensoryexperience into parts cannot be representedby linearly separable switching functions.The basic Boolean function that has thesame properties as the sign of identityis not linearly separable, and to syntheticallyrepresent linearly separability by someother means is an infinitary process.

So, we are back to Aristotle's explanationand the question of how many experiments oflength k are need to prove that two inputstrings are equivalent.

The theory of infinity in mathematics arisesbecause it is necessary if one wants tosay